TY - JOUR

T1 - Parallel randomized load balancing

T2 - A lower bound for a more general model

AU - Even, Guy

AU - Medina, Moti

PY - 2011/5/13

Y1 - 2011/5/13

N2 - We extend the lower bound of Adler et al. (1998) [1] and Berenbrink et al. (1999) [2] for parallel randomized load balancing algorithms. The setting in these asynchronous and distributed algorithms is of n balls and n bins. The algorithms begin by each ball choosing d bins independently and uniformly at random. The balls and bins communicate to determine the assignment of each ball to a bin. The goal is to minimize the maximum load, i.e., the number of balls that are assigned to the same bin. In Adler et al. (1998) [1] and Berenbrink et al. (1999) [2], a lower bound of Ω(lognloglognr) is proved if the communication is limited to r rounds. Three assumptions appear in the proofs in Adler et al. (1998) [1] and Berenbrink et al. (1999) [2]: the topological assumption, random choices of confused balls, and symmetry. The topological assumption states that each ball's decision is based only on collisions between choices of balls. The confused ball assumption states that if a ball obtains the same topological information from all its chosen bins, then the ball commits to one of the chosen bins by flipping a fair coin. The symmetry assumption states that all the balls run identical algorithms, the same assumption holds for the bins. We extend the proof of the lower bound so that it holds without these three assumptions. This lower bound applies to every parallel randomized load balancing algorithm we are aware of (Adler et al., 1998 [1]; Berenbrink et al., 1999 [2]; Stemann, 1996 [3]; Even and Medina, 2009 [4]).

AB - We extend the lower bound of Adler et al. (1998) [1] and Berenbrink et al. (1999) [2] for parallel randomized load balancing algorithms. The setting in these asynchronous and distributed algorithms is of n balls and n bins. The algorithms begin by each ball choosing d bins independently and uniformly at random. The balls and bins communicate to determine the assignment of each ball to a bin. The goal is to minimize the maximum load, i.e., the number of balls that are assigned to the same bin. In Adler et al. (1998) [1] and Berenbrink et al. (1999) [2], a lower bound of Ω(lognloglognr) is proved if the communication is limited to r rounds. Three assumptions appear in the proofs in Adler et al. (1998) [1] and Berenbrink et al. (1999) [2]: the topological assumption, random choices of confused balls, and symmetry. The topological assumption states that each ball's decision is based only on collisions between choices of balls. The confused ball assumption states that if a ball obtains the same topological information from all its chosen bins, then the ball commits to one of the chosen bins by flipping a fair coin. The symmetry assumption states that all the balls run identical algorithms, the same assumption holds for the bins. We extend the proof of the lower bound so that it holds without these three assumptions. This lower bound applies to every parallel randomized load balancing algorithm we are aware of (Adler et al., 1998 [1]; Berenbrink et al., 1999 [2]; Stemann, 1996 [3]; Even and Medina, 2009 [4]).

KW - Balls and bins

KW - Load balancing

KW - Lower bounds

KW - Static randomized parallel allocation

UR - http://www.scopus.com/inward/record.url?scp=79953293097&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2011.01.033

DO - 10.1016/j.tcs.2011.01.033

M3 - מאמר

AN - SCOPUS:79953293097

VL - 412

SP - 2398

EP - 2408

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 22

ER -